7 D-AlE 45 THE EVALUATION OF INNER PRODUCTS OF ULTIVRITE 11

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THE EVALUATIOI OF INNER PRODUCTS OFMULTIVARIATE SIMPLEX SPLINES

Thomas A. Grandine 1 ,2

Technical Summary Report #2909

February 1986

ABSTRACT

This paper gives a general method for the stable evaluation of inner products of

multivariate simplex splines. The method is based on a recurrence relation for these inner

products. The base cases for this recurrence relation are handled by triangulating certain

simploids into simplices. . . " . ,1 , - A,. ,. ,- - "7 - - ~ . (.-.

AMS(MOS) Subject Classification: 41A15, 65D07

Key Words: B-spline, simplex spline, simploid spline, multivariate, recurrence relation,linear programming, simplex method

Work Unit Number 3 - Numerical Analysis and Scientific Computing

'Sponsored by the United States Army under Contract No. DAAG29-80-C-0041.2This material is based upon work supported by the National Science Foundation under

Grant No. DMS-8210950, Mod. 4.

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SIGNIFICANCE AND EXPLANATION

For many types of projection methods (Ravleigh-Ritz-Galerkin. for example), it is

necessary to compute inner products of the basis elements in order to be able to implement %

the scheme. This paper discusses a method for evaluating these inner products in the case

where the basis elements are simplex splines.

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THE EVALUATION OF INNER PRODUCTS OFMULTIVARIATE SIMPLEX SPLINES

Thomas A. Grandine1 'H

In many practical problems, it is necessary to be able to compute the value of the

inner product of any two simplex splines. This is typically the case for many types of- - "

projection methods, including Ravleigh-Ritz-Galerkin and least squares. While various

types of quadrature methods can be used for this purpose, Dahmen and Micchelli (DM81I

report that far better results are obtained if exact values of these inner products are used.

Developing an algorithm for evaluating these inner products is the goal of this paper.

The multivariate polyhedral B-spline MB is defined as a distribution by

MBf foP, (1)R... -fe

where f is an arbitrary m-variate function, B is a polyhedral body in R" , n > m, and

P R' -- R m is a linear map. Typically, P is chosen to be the canonical projector, i.e.

P x x z(i)",. This is a generalization of an earlier definition due to Micchelli in which

B is chosen specifically to be a simplex. In particular, if x0, xi, ... , x, are points in R",

and if IA? denotes the convex hull of A, then the multivariate simplex spline M(. x, X ,,)-

is defined to be [M801:

M( X1 ,/,x)f f oOP. (2)

'Sponsored by the United States Army under Contract No. DAAG29-80-C-0041.2This material is based upon work supported by the National Science Foundation underGrant No. DMS-8210950. Mod. 4.

.-."-....-..-..-.-. ................................ ..... . ... ,-

- -.. . . . . . . . . . .. . . . . . . . . . . . ..

' . 7. . . .

Lii

This definition, which is a convenient reformulation of an earlier definition of de Boor

!B76, is a special case of definition (1) except for the difference in normalization.

The purpose of this paper is to describe a method for evaluating inner products of

simplex splines (with the normalization given in (2)). The inner product referred to here

is the one given by the following integral:

(f, g) fg.

An identity can be used to evaluate the integral in the case where I and g happen to be

two polyhedral splines, say MB and M-, defined as in (1) byf MB! B 'oP * "

MBf :=

for two ordinary linear maps P and Q into R m . From [DM82-2], it is known that the

convolution of two polyhedral splines satisfies

M C() MB(Y - )Mc(y)dy, (3)

where B C is the polyhedral set which is the usual Cartesian product of the polyhedral

sets B and C, i.e. B x C := {(r,s)Ir E B,s E C}. Setting x = 0 in this equation results

in the identity

J" M,(x)Mc(x)dx MB>(c(O), (4)

where

J MB (x)f(x)dx fI (y- Px)dydx. (5)

If B and C are simplices. then this relation says that the value of the inner product of two

simplex splines is given by the value of a certain simploid spline at the origin. A simploid

2

2.. . .. . . . . .". . . . . . .. ... -

is defined as the Cartesian product of two simplices. Thus, if an accurate way can be

found to evaluate this simploid spline, then an accurate way has been found to compute

the value of the desired inner product.

Dahmen and Micchelli !DM81-2! have proposed the use of the recurrence relation for

polyhedral splines for this purpose. Specialized to the case of the simploid, and then

rewritten in terms of inner products, this recurrence relation takes on the following form

IDM81-21:

Theorem 1: If B = lxo,...,xl, C = [y,.-., Yp, and Y'"oaixt = ojyj, with

-o a= _:o/ = 1, then

M(zlxo,".,x)M(zjY",...,y)dx = 1 pn +p- m

n

M (X1X '---' i-1'(6)

p L j MwzXo, .... 9Xn)M(x yo,...yj-1, yj+1, ... , ypldx• "

This recurrence relation can be implemented using the linear programming approach.'V,.4.

of [G841. This technique requires, at the first step, a solution to the following problem:

Find a, and 3 so that

"n Patz, - L'jj 0

j=0

ai

o(7)o -.. -). , 17))- "---

a, 0, i=O,...,n

0o, 0o .,

....... I13

. .".. . .......... . . .. . . . . o o .- -. , ,..- -. o . . . 0o .. • . .....- -. .o . . - ° . • 0 . . .. . -

.' ." • , , .•.. " . - , . - .-. * - .o .* , ... ". .". " ... .•,- -.. .- . . . - . . . . . -o . .- . ." . - . .- .

As described in G84:. problems of this sort can be s. .', via the dual simplex method for

linear programming problems. This method always results in solutions for which all but

m -. 2 of the a, and 3, are zero. This is because the problem (7) has only m + 2 equality

constraints in it. and solutions to linear programming problems can always be made to- p"

satisfy a cornplementarity condition (see 1Da63l and Ma78'). This number is a drastic

improvement over the 2m -- 2 given by Dahmen and Micchelli in IDM8IJ.

In any case, the implementation of the recurrence relation (6) proceeds just as in 1G841.

The snag appears when one or the other (or both) of the simplex splines appearing on the

left-hand-side of (6) are piecewise constant functions, which happens when a simplex spline-p

has exactly m + 1 knots present. In this case, the recurrence relation cannot be applied,

for then some of the functions appearing on the right-hand-side of the recurrence relation

will not be well-defined. Dahmen and Micchelli conquer this obstacle by expressing each

of the simplex splines in the integral on the left-hand-side of (6) as a linear combination

of cone splines. Then, since the convolution of two cone splines is again a cone spline,

they are able to write this inner product as a linear combination of cone splines (see [D79],

ID801, and IDM81-2]), each of which may be evaluated using the recurrence relation for

cone splines.

There is a way, however, to avoid going to the trouble of implementing the recurrence

relation for cone splines. In fact, the cone splines can be dispensed with altogether by not-

ing, as was done already in [H821 and IDM821, that any simploid can easily be triangulated i.

merely by arranging its vertices in a rectangular grid and tracing the various paths through

this grid. This makes it possible to express any simploid spline as a linear combination of

simplex splines, each of which may be evaluated using the method described in 1G841.

4

.........~~~~~~.......:..-..i..... ................. . . . . .

In what follows, let Ei := 'Zo,...,x,,] and E2 Yo,...,y,4. With t ( ), let

oT..... at be a collection of simplices such that {oa} is the triangulation of E, X E2 that is

produced via the following construction ]H82.: Consider the following grid, representing

the set E, x 2,(Xo,YP) (xI,Y,) ... (Xn,Y)

(Xo, YO) (X, YO).. (xn,YO)" ",XY ) ( ... (.Yo

Take all the paths through this grid given by a = {so, ... S;,}, where so = (xo,yo),

sn+ : = (x:,y,), and if s, = (Zp,ye), then si-, must be either (xi,y) or (x,,y-,)..

There are (n) different paths through the grid. It can be shown that the set of all paths

through the grid is a triangulation of the simploid (Lemma 3 in 1H821).

This construction ensures the truth of

Lemma 1: .-. '

voIE X E2voli= o-

A. .-

Proofr Let Pj be the canonical projector from E X E into Ej. Then..

volEA X E2 volE(, IsV-(n + p)!

nvoM(PXox.. . Xn)M(XIY(.. . yF)dx (It) volo, (8)

5S .

Z.Ig

, - -r...,M ,.1,,. ,,. .-. o . ,,= . . 3 .- , -' .

4-

Proof: By definition,

ME, r". = f P

fo

so that ME, . , M,,. Then

I~M I" "°E

M(xjxo ....,,)M(xjy( ..,yp)dx = V-lE1V....dx

-R volIx~E E 2 (O= ME, x E2 (0) -:

volIE x E 2

t1 ___(O _"",". " ".

volE 1 X E2 ±v" .oMaio (0)_ vlo vuo ""

As it turns out, the terms on the right-hand-side of this formula are all normalized

in the standard way for simplex splines. Hence, the simplex splines appearing on the

right-hand-side of (8) are those obtained by considering all possible paths through the

grid:XOjYp XI Yp.. n Y

xvoy i X1 -Y ... X'"

xco YOXi-Yo ... Xn Yo

-----

A nFormula (8) provides an easy way of evaluating inner products when the recurrence .".

relation (6) cannot be used because one or more of the simplex splines in the integral

have only m - I knots. It also provides an easy way to dispense with (6) altogether, since

(8) is tre independent of the number of knots appearing in the simplex splines in the

6

V ...

LIP",]-y .. z,- ] -"..

integral. As elegant and clean as this approach is, however, it is much less efficient than ..

using recurrence relation (6) whenever possible, and then using relation (8) the rest of the

time.

This is best seen by considering the following calculation. Suppose that the value of

an inner product of a simplex spline with n + 1 knots and a simplex spline of p + 1 knots

is desired. To obtain this value using formula (8) will require the evaluation of ( n")

simplex splines of degree n + p - m. Each of these simplex splines requires m 4 1 times as

much work to evaluate as does a simplex spline of degree n + p - m - 1. Thus, the total

amount of work required is roughly equivalent to that of evaluating (m + 1) (n+p) simplex

splines of degree n + p - m - 1. Now consider what happens when one application of the

recurrence relation (6) is applied, followed by this triangulation technique. The recurrence

relation will produce a sum of inner products, say b, inner products of n + 1 knot and

p knot simplex splines, and 62 inner products of n knot and p + I knot simplex splines.

Here b, + b2 = m 2, as remarked earlier. After triangulating each of these simploids,

the total number of degree n + p - m - I simplex splines which need to be evaluated is

61 (n+ -) + 62 Since b, <_ m+ I and b2 _ m+ 1, with equality in at most

one of them, it follows that 6'p + 62n < (m + 1)(n + p). Multiplying this inequality by

(n + p- 1)!/(n!p!) yields

(1 + (b2 < (m) + (n (9)

Thus, it is always more efficient to employ recurrence relation (6) whenever that is possible.

Example: Suppose that n p 5 and m 2 above. Then using the recurrence

relation (6) once requires evaluation of 504 simplex splines of degree 7, while not using it

7

-6 %

at all requires evaluation of 756 simplex spiii-ie of de-?r~ 7. lialf agair. as much work. Of

course, itwould mnake sense to employ (6) as often as possible to copt inner prdcs

The point of this example is that even one application can make a tremendous difference.

.4o

b"* Z4

* References

B76, de Boor, Carl. "Splines as linear combinations of B-splines," in Approximation Theory

II, G. G. Lorentz. C. K. Chui, and L. L. Schumaker, eds., Academic Press, pp. 1-47.

*i B821 de Boor, Carl. "Topics in Multivariate Approximation Theory," in Topics in Numerical

Analysis, P. Turner, ed. Springer Lecture Notes in Mathematics 965, 1982, pp. 39-78.

'BH811 de Boor, Carl, and H6llig, Klaus. "Recurrence Relations for Multivariate B-splines,"

Proc. Amer. Math. Soc. 85, 1982, pp. 397-400.

* D791 Dahmen, W. "Multivariate B-splines - Recurrence Relations and Linear Combinations

of Truncated Powers," in Multivariate Approximation Theory, W. Schempp and K.

Zeller, eds., Basel: Birkhfiuser, pp. 64-82.

[ DM8lj Dahmen, W., and Micchelli, C. A. "Numerical Algorithms for Least Squares Approx-

imation by Multivariate B-splines," in Numerical Methods of Approximation Theory,

Vol. 6, ed. by Collatz, Meinardus, and Werner, Basel: Birkhiuser, 1981.

D.%81-2' Dahmen, W., and Micchelli, C. A. "Computation of Inner Products of Multivariate

B-splines$ Nurner. Funet. Anal. and Optimiz. 3(3), pp. 367-375.

DM821 Dahmen, W., and Micchelli, C. A. "On the Linear Independence of Multivariate B-

splines I., Triangulation of Simploids," SIAM Journal of Numerical Analysi's 19, pp.

993-1012.

9

%.-

n -WN W- ! . . .- ,- u .

V.'2-',: Dahnien, W.. and Micchvllj. C. A. "Some Remarks on Multivariate B-splines," in

Muittmariate Approximation Theory 11, W. Schemp, K. Zeller. eds. Basel: Birkhiuser,

1.82. pp. 81-87.

,,:; l)altzig, G. B. Linear Programming and Extensions. Princeton University Press, 1963.

* (>. 1 Grandine. Thornwa. "The Stable Evaluation of Multivariate B-splines," Mathematics .

Research Center TSR .2744, 1984.

--" H6lig. Klaus. "Muiltivariate Splines," SIAM Journal of Numerical Analysis 19, pp. ,

1013 1031.

- , iangasarian, Olvi. "Linear Programming Lecture Notes." Manuscript, 1978.

"1 40 Micchelli, C. A. "A Constructive Approach to Kergin Interpolation in Rk: Multivariate

B-splines and Lagrange Interpolation," Rocky Mountains J. Math. 10, pp. 485-497.

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